SECTION 8 Newto s Method 9 (a) a a Sectio 8 (, ( )) (, ( )) Taget lie c Taget lie c b (b) The -itercept o the taget lie approimates the zero o Figure 60 b Newto s Method Approimate a zero o a uctio usig Newto s Method Newto s Method I this sectio ou will stud a techique or approimatig the real zeros o a uctio The techique is called Newto's Method, ad it uses taget lies to approimate the graph o the uctio ear its -itercepts To see how Newto s Method works, cosider a uctio that is cotiuous o the iterval a, b ad dieretiable o the iterval a, b I a ad b dier i sig, the, b the Itermediate Value Theorem, must have at least oe zero i the iterval a, b Suppose ou estimate this zero to occur at First estimate as show i Figure 60(a) Newto s Method is based o the assumptio that the graph o ad the taget lie at, both cross the -ais at about the same poit Because ou ca easil calculate the -itercept or this taget lie, ou ca use it as a secod (ad, usuall, better) estimate or the zero o The taget lie passes through the poit, with a slope o I poit-slope orm, the equatio o the taget lie is thereore Lettig 0 ad solvig or produces So, rom the iitial estimate You ca improve o ou obtai a ew estimate Secod estimate [see Figure 60(b)] ad calculate et a third estimate Third estimate Repeated applicatio o this process is called Newto s Method Video NEWTON S METHOD Isaac Newto irst described the method or approimatig the real zeros o a uctio i his tet Method o Fluios Although the book was writte i 67, it was ot published util 76 Meawhile, i 690, Joseph Raphso (648 75) published a paper describig a method or approimatig the real zeros o a uctio that was ver similar to Newto s For this reaso, the method is ote reerred to as the Newto- Raphso method Newto s Method or Approimatig the Zeros o a Fuctio Let c 0, where is dieretiable o a ope iterval cotaiig c The, to approimate c, use the ollowig steps Make a iitial estimate that is close to c (A graph is helpul) Determie a ew approimatio I is withi the desired accurac, let serve as the ial approimatio Otherwise, retur to Step ad calculate a ew approimatio Each successive applicatio o this procedure is called a iteratio Techolog
0 CHAPTER Applicatios o Dieretiatio NOTE For ma uctios, just a ew iteratios o Newto s Method will produce approimatios havig ver small errors, as show i Eample EXAMPLE Usig Newto s Method Calculate three iteratios o Newto s Method to approimate a zero o Use as the iitial guess Solutio Because, ou have, ad the iterative process is give b the ormula = = 5 The calculatios or three iteratios are show i the table () = 000000 000000 000000 0500000 500000 500000 050000 000000 008 46667 46667 0006945 84 00045 446 4 446 The irst iteratio o Newto s Method Figure 6 O course, i this case ou kow that the two zeros o the uctio are ± To si decimal places, 444 So, ater ol three iteratios o Newto s Method, ou have obtaied a approimatio that is withi 000000 o a actual root The irst iteratio o this process is show i Figure 6 Editable Graph Tr It Eploratio A Eploratio B EXAMPLE Usig Newto s Method () = + + Use Newto s Method to approimate the zeros o Cotiue the iteratios util two successive approimatios dier b less tha 0000 Solutio Begi b sketchig a graph o, as show i Figure 6 From the graph, ou ca observe that the uctio has ol oe zero, which occurs ear Net, dieretiate ad orm the iterative ormula 6 The calculatios are show i the table Ater three iteratios o Newto s Method, the zero o is approimated to the desired accurac Figure 6 Editable Graph 0000 08400 54000 005 5 00077 56876 0006 75 00000 5665 000000 4 75 5 75 75 Because two successive approimatios dier b less tha the required 0000, ou ca estimate the zero o to be 75 Tr It Eploratio A Eploratio B Ope Eploratio Video
SECTION 8 Newto s Method Whe, as i Eamples ad, the approimatios approach a limit, the sequece,,,,, is said to coverge Moreover, i the limit is c, it ca be show that c must be a zero o Newto s Method does ot alwas ield a coverget sequece Oe wa it ca ail to do so is show i Figure 6 Because Newto s Method ivolves divisio b, it is clear that the method will ail i the derivative is zero or a i the sequece Whe ou ecouter this problem, ou ca usuall overcome it b choosig a dieret value or Aother wa Newto s Method ca ail is show i the et eample ( ) = 0 Newto s Method ails to coverge i Figure 6 0 EXAMPLE A Eample i Which Newto s Method Fails () = / The uctio is ot dieretiable at 0 Show that Newto s Method ails to coverge usig 0 Solutio Because, the iterative ormula is The calculatios are show i the table This table ad Figure 64 idicate that cotiues to icrease i magitude as, ad so the limit o the sequece does ot eist 4 5 Newto s Method ails to coverge or ever -value other tha the actual zero o Figure 64 00000 04646 5470 00000 00000 00000 058480 097467 060000 040000 040000 0768 0640 0000 080000 4 080000 098 08680 40000 60000 Tr It Eploratio A Eploratio B NOTE I Eample, the iitial estimate 0 ails to produce a coverget sequece Tr showig that Newto s Method also ails or ever other choice o (other tha the actual zero)
CHAPTER Applicatios o Dieretiatio It ca be show that a coditio suiciet to produce covergece o Newto s Method to a zero o is that < Coditio or covergece o a ope iterval cotaiig the zero For istace, i Eample this test would ield,,, ad Eample 4 O the iterval,, this quatit is less tha ad thereore the covergece o Newto s Method is guarateed O the other had, i Eample, ou have,, 9 5, ad Eample 9 5 9 4 which is ot less tha or a value o, so ou caot coclude that Newto s Method will coverge Algebraic Solutios o Polomial Equatios The zeros o some uctios, such as ca be oud b simple algebraic techiques, such as actorig The zeros o other uctios, such as caot be oud b elemetar algebraic methods This particular uctio has ol oe real zero, ad b usig more advaced algebraic techiques ou ca determie the zero to be 6 6 Because the eact solutio is writte i terms o square roots ad cube roots, it is called a solutio b radicals NIELS HENRIK ABEL (80 89) MathBio EVARISTE GALOIS (8 8) MathBio Although the lives o both Abel ad Galois were brie, their work i the ields o aalsis ad abstract algebra was ar-reachig NOTE Tr approimatig the real zero o ad compare our result with the eact solutio show above The determiatio o radical solutios o a polomial equatio is oe o the udametal problems o algebra The earliest such result is the Quadratic Formula, which dates back at least to Babloia times The geeral ormula or the zeros o a cubic uctio was developed much later I the siteeth cetur a Italia mathematicia, Jerome Carda, published a method or idig radical solutios to cubic ad quartic equatios The, or 00 ears, the problem o idig a geeral quitic ormula remaied ope Fiall, i the ieteeth cetur, the problem was aswered idepedetl b two oug mathematicias Niels Herik Abel, a Norwegia mathematicia, ad Evariste Galois, a Frech mathematicia, proved that it is ot possible to solve a geeral ith- (or higher-) degree polomial equatio b radicals O course, ou ca solve particular ith-degree equatios such as 5 0, but Abel ad Galois were able to show that o geeral radical solutio eists
SECTION 8 Newto s Method Eercises or Sectio 8 The smbol Click o Click o idicates a eercise i which ou are istructed to use graphig techolog or a smbolic computer algebra sstem to view the complete solutio o the eercise to prit a elarged cop o the graph I Eercises 4, complete two iteratios o Newto s Method or the uctio usig the give iitial guess, 7, si, 4 ta, 0 I Eercises 5 4, approimate the zero(s) o the uctio Use Newto s Method ad cotiue the process util two successive approimatios dier b less tha 000 The id the zero(s) usig a graphig utilit ad compare the results 5 6 5 7 8 9 0 9 479 88 4 si 4 cos (a) Use Newto s Method ad the uctio a to derive the Mechaic s Rule (b) Use the Mechaic s Rule to approimate 5 ad 7 to three decimal places 0 (a) Use Newto s Method ad the uctio a to obtai a geeral rule or approimatig a (b) Use the geeral rule oud i part (a) to approimate 4 6 ad 5 to three decimal places I Eercises 4, appl Newto s Method usig the give iitial guess, ad eplai wh the method ails 6 6, 4, I Eercises 5 8, appl Newto s Method to approimate the -value(s) o the idicated poit(s) o itersectio o the two graphs Cotiue the process util two successive approimatios dier b less tha 000 [Hit: Let h g ] 5 6 g 4 g g 4 Figure or Figure or 6 0 6, si cos, g π π 7 8 g ta g cos Figure or Figure or 4 6 4 g π π π π g Writig About Cocepts 5 I our ow words ad usig a sketch, describe Newto s Method or approimatig the zeros o a uctio 6 Uder what coditios will Newto s Method ail? 9 Mechaic s Rule The Mechaic s Rule or approimatig a, a > 0, is a, where,, is a approimatio o a Fied Poit I Eercises 7 ad 8, approimate the ied poit o the uctio to two decimal places [ A ied poit 0 o a uctio is a value o such that 0 0 ] 7 cos 8 ) cot, 0 < <
4 CHAPTER Applicatios o Dieretiatio 9 Writig Cosider the uctio (a) Use a graphig utilit to graph (b) Use Newto s Method with as a iitial guess (c) Repeat part (b) usig 4 as a iitial guess ad observe that the result is dieret (d) To uderstad wh the results i parts (b) ad (c) are dieret, sketch the taget lies to the graph o at the poits, ad 4, 4 Fid the -itercept o each taget lie ad compare the itercepts with the irst iteratio o Newto s Method usig the respective iitial guesses (e) Write a short paragraph summarizig how Newto s Method works Use the results o this eercise to describe wh it is importat to select the iitial guess careull 0 Writig Repeat the steps i Eercise 9 or the uctio si with iitial guesses o 8 ad Use Newto s Method to show that the equatio a ca be used to approimate a i is a iitial guess o the reciprocal o a Note that this method o approimatig reciprocals uses ol the operatios o multiplicatio ad subtractio [Hit: Cosider a ] Use the result o Eercise to approimate (a) ad (b) to three decimal places I Eercises ad 4, approimate the critical umber o o the iterval 0, Sketch the graph o, labelig a etrema cos 4 si I Eercises 5 8, some tpical problems rom the previous sectios o this chapter are give I each case, use Newto s Method to approimate the solutio 5 Miimum Distace Fid the poit o the graph o 4 that is closest to the poit, 0 6 Miimum Distace Fid the poit o the graph o that is closest to the poit 4, 7 Miimum Time You are i a boat miles rom the earest poit o the coast (see igure) You are to go to a poit Q, which is miles dow the coast ad mile ilad You ca row at miles per hour ad walk at 4 miles per hour Toward what poit o the coast should ou row i order to reach Q i the least time? mi mi 8 Medicie The cocetratio C o a chemical i the bloodstream t hours ater ijectio ito muscle tissue is give b C t t 50 t Whe is the cocetratio greatest? Q mi 9 Advertisig Costs A compa that produces portable CD plaers estimates that the proit or sellig a particular model is P 76 480 0,000, where P is the proit i dollars ad is the advertisig epese i 0,000s o dollars (see igure) Accordig to this model, id the smaller o two advertisig amouts that ield a proit P o $,500,000 Proit (i dollars),000,000,000,000,000,000 P 0 0 50 Advertisig epese (i 0,000s o dollars) 90 80 70 60 50 40 0 0 Figure or 9 Figure or 40 40 Egie Power The torque produced b a compact automobile egie is approimated b the model T 0808 7974 748 084, 4 5 Egie speed (i thousads o rpm) 5 where T is the torque i oot-pouds ad is the egie speed i thousads o revolutios per miute (see igure) Approimate the two egie speeds that ield a torque T o 70 oot-pouds True or False? I Eercises 4 44, determie whether the statemet is true or alse I it is alse, eplai wh or give a eample that shows it is alse 4 The zeros o p q coicide with the zeros o p 4 I the coeiciets o a polomial uctio are all positive, the the polomial has o positive zeros 4 I is a cubic polomial such that is ever zero, the a iitial guess will orce Newto s Method to coverge to the zero o 44 The roots o 0 coicide with the roots o 0 45 Taget Lies The graph o si has iiitel ma taget lies that pass through the origi Use Newto s Method to approimate the slope o the taget lie havig the greatest slope to three decimal places 46 Cosider the uctio 0 4 (a) Use a graphig utilit to determie the umber o zeros o (b) Use Newto s Method with a iitial estimate o to approimate the zero o to our decimal places (c) Repeat part (b) usig iitial estimates o 0 ad 00 (d) Discuss the results o parts (b) ad (c) What ca ou coclude? Torque (i t-lbs) 0 60 T